Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 6, 1-14; http://www.math.u-szeged.hu/ejqtde/

On the stability of some fractional-order non-autonomous systems

Sheren A. Abd El-Salam e.mail: shrnahmed@maktoob.com Ahmed M. A. El-Sayed e.mail: amasayed@hotmail.com Faculty of Science, Alexandria University, Alexandria, Egypt

Abstract

The fractional calculus (integration and differentiation of fractional-order) is a one of the singular integral and integro-differential operators. In this work a class of fractionalorder non-autonomous systems will be considered. The stability (and some other properties concerning the existence and uniqueness) of the solution will be proved.

Key words: Fractional calculus, fractional-order non-autonomous systems, stability, asymptotic stability.

1 Introduction

Let L_{1}[a, b] denotes the space of all Lebesgue integrable functions on the interval [a, b], 0 ≤ a < b < ∞. Definition 1.1 The fractional (arbitrary) order integral of the function f ∈ L_{1}[a, b] of order β ∈ R^{+} is defined by (see [2] and [4] - [6])

I_{a}^{β} f(t) = ∫_{a}^{t} \frac{(t − s)^{β − 1}}{Γ(β)} f(s) ds,

where Γ(.) is the gamma function. Definition 1.2 The Riemann-Liouville fractional-order derivative of f(t) of order α ∈ (0, 1) is defined as (see [2] and [4] - [6])

∗D_{a}^{α} f(t) = \frac{d}{dt} I_{a}^{1 − α} f(t), t ∈ [a, b].

Definition 1.3 The (Caputo) fractional-order derivative D^{α} of order α ∈ (0, 1] of the function g(t) is defined as (see [4] - [6])

D_{a}^{α} g(t) = I_{a}^{1 − α} \frac{d}{dt} g(t), t ∈ [a, b].

Now consider the non-autonomous linear system:

x′(t) = A(t) x(t),  (1)

with the initial condition

x(t_{0}) = x^{0}, t ≥ t_{0},

where A(t) is a continuous n by n matrix on the half-axis t ≥ 0. We know that (see [1]) the solution of system (1) is given by:

x(t, t_{0}, x^{0}) = X(t) X^{−1}(t_{0}) x^{0}, t ≥ 0,

where X(t) is an arbitrary fundamental matrix of the system (1) defined on the whole half-axis t ≥ 0. Now we shall present the main definitions (see [1]) related to the concepts of stability of the solution x = 0 of (1). Definition 1.4 The solution x = 0 of (1) will be called stable if to any ε > 0, t_{0} ≥ 0 there corresponds δ(ε, t_{0}) > 0 such that ||x(t, t_{0}, x^{0})|| < ε for t ≥ t_{0} as soon as ||x^{0}|| < δ. Definition 1.5 The solution x = 0 of (1) will be called uniformly stable if δ(ε, t_{0}) from definition 1.4 can be chosen independent of t_{0} : δ(ε, t_{0}) ≡ δ(ε). Definition 1.6 The solution x = 0 of (1) will be called asymptotically stable if it is stable in the sense of definition 1.4 and there exists γ(t_{0}) > 0 such that lim_{t→∞} ||x(t, t_{0}, x^{0})|| = 0 for

every x(t, t_{0}, x^{0}) with ||x^{0}|| < γ. Definition 1.7 The solution x = 0 of (1) will be called uniformly asymptotically stable if it is uniformly stable in the sense of definition 1.5 and moreover, for any ε > 0 there exists T (ε) > 0 such that ||x(t, t_{0}, x^{0})|| < ε for every t ≥ t_{0} + T (ε) and all x^{0} with ||x^{0}|| < γ_{0}, where γ_{0} is independent of t_{0}. In other words, x = 0 of (1) will be called uniformly asymptotically stable if it is uniformly stable and lim||x(t, t_{0}, x^{0})|| = 0 as t − t_{0} → +∞ uniformly with respect to (t_{0}, x^{0}), t_{0} ≥ 0, ||x^{0}|| < γ_{0}.

Theorem 1.1

Let X(t) be a fundamental matrix of the system (1). A necessary and sufficient condition for the stability of the solution x = 0 is the boundedness of X(t) on t ≥ 0:

||X(t)|| ≤ M, t ≥ 0.

A necessary and sufficient condition for the asymptotic stability of the solution x = 0 is

lim_{t→∞} ||X(t)|| = 0.

Theorem 1.2

Let X(t) be a fundamental matrix of the system (1). A necessary and sufficient condition for the uniform stability of the solution x = 0 is the existence of a number M > 0 such that:

||X(t) X^{−1}(t_{0})|| ≤ M, t ≥ t_{0} ≥ 0.

A necessary and sufficient condition for the uniform asymptotic stability of the solution x = 0 is the existence of two positive numbers M and η such that:

||X(t) X^{−1}(t_{0})|| ≤ M exp[−η (t − t_{0})], t ≥ t_{0} ≥ 0.

Here in this work, we study the stability (and some other properties concerning the existence and uniqueness) of the solutions of the non-autonomous linear systems:

D_{t_{0}}^{α} x(t) = A(t) x(t) + f(t), α ∈ (0, 1]

and

x′(t) = A(t) \frac{d}{dt} I_{t_{0}}^{α} x(t) + f(t), α ∈ (0, 1],

with the initial condition

x(t_{0}) = x^{0}.

Also the special cases:

D_{t_{0}}^{α} x(t) = A x(t), α ∈ (0, 1], x(t_{0}) = x^{0}

and

x′(t) = A \frac{d}{dt} I_{t_{0}}^{α} x(t), α ∈ (0, 1], x(t_{0}) = x^{0}

will be studied.

2 Existence of solution

Here the space B[t_{0}, T] denotes the space of all n vector functions y such that e^{−Nt}|y_{i}(t)| ∈ L_{1}[t_{0}, T], T < ∞, N > 0, and the space C^{∗}[t_{0}, T] denotes the space of all n vector functions x such that e^{−Nt}|x_{i}(t)| ∈ C[t_{0}, T], T < ∞, N > 0, while the space AC^{∗}[t_{0}, T] denotes the space of all n vector absolutely continuous functions, in addition the norm on B[t_{0}, T] will be denoted by ||.||_{1}, that is, for y ∈ B[t_{0}, T], ||y||_{1} = ∑_{i=1}^{n} ||y_{i}||_{1} = ∑_{i=1}^{n} ∫_{t_{0}}^{t} e^{−Ns}|y_{i}(s)| ds, while the norm on C^{∗}[t_{0}, T] will be denoted by ||.||_{2}, that is, for x ∈ C^{∗}[t_{0}, T], ||x||_{2} = ∑_{i=1}^{n} ||x_{i}||_{2} = ∑_{i=1}^{n} sup_{t} e^{−Nt}|x_{i}(t)|. Throughout this paper we define an n × n matrix function A(t) = (a_{ij}(t)), i, j = 1, 2, ..., n such that A : [t_{0}, T] → R, T < ∞, also define

||A^{∗}|| := ||a_{ij}^{∗}|| = sup_{t} |a_{ij}(t)| and ||Ã|| := ||ã_{ij}|| = sup_{t} |a′_{ij}(t)|

Consider firstly the problem:

D_{t_{0}}^{α} x(t) = A(t) x(t) + f(t), α ∈ (0, 1], x(t_{0}) = x^{0}.  (2)

Theorem 2.1 Let f(t) ∈ AC^{∗}[t_{0}, T]. If A(t) ∈ AC^{∗}[t_{0}, T], then there exists a unique solution of problem (2). Proof. Problem (2) is equivalent to the equation

y(t) = x^{0} \frac{d}{dt} I_{t_{0}}^{α} A(t) + \frac{d}{dt} I_{t_{0}}^{α} A(t) I y(t) + \frac{d}{dt} I_{t_{0}}^{α} f(t).  (3)

Indeed: let x(t) be a solution of (2) and take y(t) = x′(t) ⇒ x(t) = x^{0} + I y(t), then

I_{t_{0}}^{1 − α} y(t) = A(t) (x^{0} + I y(t)) + f(t)

Operating by I_{t_{0}}^{α} on both sides of the last equation, we obtain

I y(t) = x^{0} I_{t_{0}}^{α} A(t) + I_{t_{0}}^{α} A(t) I y(t) + I_{t_{0}}^{α} f(t),

differentiating both sides, we get (3). Conversely Let y(t) be a solution of (3), take y(t) = x′(t) ⇒ x(t) = x^{0} + I y(t) and x(t_{0}) = x^{0}, then

x′(t) = x^{0} \frac{d}{dt} I_{t_{0}}^{α} A(t) + \frac{d}{dt} I_{t_{0}}^{α} A(t) (x(t) − x^{0}) + \frac{d}{dt} I_{t_{0}}^{α} f(t)
= \frac{d}{dt} I_{t_{0}}^{α} A(t) x(t) + \frac{d}{dt} I_{t_{0}}^{α} f(t).

Operating by I_{t_{0}}^{1 − α} on both sides of the last equation, we obtain

I_{t_{0}}^{1 − α} x′(t) = I_{t_{0}}^{1 − α} \frac{d}{dt} I_{t_{0}}^{α} A(t) x(t) + I_{t_{0}}^{1 − α} \frac{d}{dt} I_{t_{0}}^{α} f(t)
= I_{t_{0}}^{1 − α} (A(t_{0}) x^{0} \frac{(t − t_{0})^{α−1}}{Γ(α)} + I_{t_{0}}^{α} (A(t) x(t))′) + \frac{d}{dt} I_{t_{0}}^{1 − α} I_{t_{0}}^{α} f(t)
= A(t_{0}) x^{0} + A(t) x(t) − A(t_{0}) x^{0} + f(t),

then

D_{t_{0}}^{α} x(t) = A(t) x(t) + f(t).

Which proves the equivalence. Now define the operator F : B → B by

F y(t) = x^{0} \frac{d}{dt} I_{t_{0}}^{α} A(t) + \frac{d}{dt} I_{t_{0}}^{α} A(t) I y(t) + \frac{d}{dt} I_{t_{0}}^{α} f(t).  (4)

Let y_{i}, z_{i} ∈ B, then

F y_{i}(t) − F z_{i}(t) = \frac{d}{dt} I_{t_{0}}^{α} a_{ij}(t) I (y_{j}(t) − z_{j}(t))
= I_{t_{0}}^{α} a′_{ij}(t) I (y_{j}(t) − z_{j}(t)) + I_{t_{0}}^{α} a_{ij}(t) (y_{j}(t) − z_{j}(t)),
e^{−Nt} |F y_{i}(t) − F z_{i}(t)| ≤ e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α − 1}}{Γ(α)} |a′_{ij}(s)| ∫_{t_{0}}^{s} |y_{j}(θ) − z_{j}(θ)| dθ ds
+ e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α − 1}}{Γ(α)} |a_{ij}(s)| |y_{j}(s) − z_{j}(s)| ds

≤ ||ã_{ij}|| e^{−Nt} ∫_{t_{0}}^{t} |y_{j}(θ) − z_{j}(θ)| ∫_{θ}^{t} \frac{(t − s)^{α − 1}}{Γ(α)} ds dθ
+ ||a_{ij}^{∗}|| e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α − 1}}{Γ(α)} |y_{j}(s) − z_{j}(s)| ds
≤ ||ã_{ij}|| e^{−Nt} ∫_{t_{0}}^{t} |y_{j}(θ) − z_{j}(θ)| \frac{(t − θ)^{α}}{Γ(1 + α)} dθ
+ ||a_{ij}^{∗}|| e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α − 1}}{Γ(α)} |y_{j}(s) − z_{j}(s)| ds,

and

∫_{t_{0}}^{t} e^{−Ns} |F y_{i}(s) − F z_{i}(s)| ds ≤ ||ã_{ij}|| ∫_{t_{0}}^{t} e^{−Ns} ∫_{t_{0}}^{s} |y_{j}(θ) − z_{j}(θ)| \frac{(s − θ)^{α}}{Γ(1 + α)} dθ ds
+ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Ns} ∫_{t_{0}}^{s} \frac{(s − θ)^{α − 1}}{Γ(α)} |y_{j}(θ) − z_{j}(θ)| dθ ds
≤ ||ã_{ij}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − z_{j}(θ)| ∫_{θ}^{t} e^{−N(s−θ)} \frac{(s − θ)^{α}}{Γ(1 + α)} ds dθ
+ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − z_{j}(θ)| ∫_{θ}^{t} e^{−N(s−θ)} \frac{(s − θ)^{α−1}}{Γ(α)} ds dθ
≤ ||ã_{ij}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − z_{j}(θ)| ∫_{0}^{N(t−θ)} e^{−u} \frac{u^{α}}{N^{α} Γ(1 + α)} \frac{du}{N} dθ
+ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − z_{j}(θ)| ∫_{0}^{N(t−θ)} e^{−u} \frac{u^{α−1}}{N^{α−1} Γ(α)} \frac{du}{N} dθ
≤ \frac{||ã_{ij}||}{N^{1+α}} ||y_{j} − z_{j}||_{1} + \frac{||a_{ij}^{∗}||}{N^{α}} ||y_{j} − z_{j}||_{1}
≤ (\frac{||ã_{ij}||}{N^{1+α}} + \frac{||a_{ij}^{∗}||}{N^{α}}) ||y_{j} − z_{j}||_{1},

therefore

||F y_{i} − F z_{i}||_{1} ≤ (\frac{||ã_{ij}||}{N^{1+α}} + \frac{||a_{ij}^{∗}||}{N^{α}}) ||y_{j} − z_{j}||_{1},

∑_{i=1}^{n} ||F y_{i} − F z_{i}||_{1} ≤ ∑_{i=1}^{n} (\frac{||ã_{ij}||}{N^{1+α}} + \frac{||a_{ij}^{∗}||}{N^{α}}) ||y_{j} − z_{j}||_{1},

||F y − F z||_{1} ≤ (\frac{||Ã||}{N^{1+α}} + \frac{||A^{∗}||}{N^{α}}) ||y − z||_{1}.

Now choose N large enough such that \frac{||_{A}||}{N^{1+α}} + \frac{||A^{∗}||}{N^{α}} < 1, then we get

||F y − F z||_{1} < ||y − z||_{1},

therefore the map F : B → B is contraction and (4) has a unique fixed point y ∈ B[t_{0}, T], therefore we deduce that the problem (2) has a unique solution x ∈ AC^{∗}[t_{0}, T].

Consider secondly the problem

x′(t) = A(t) \frac{d}{dt} I_{t_{0}}^{α} x(t) + f(t), α ∈ (0, 1], x(t_{0}) = x^{0}.  (5)

Theorem 2.2

Let f(t) ∈ B[t_{0}, T]. If A(t) be an n × n matrix function which is bounded and measurable, then there exists a unique solution of problem (5). Proof. Problem (5) is equivalent to the equation

y(t) = x^{0} \frac{(t − t_{0})^{α − 1}}{Γ(α)} A(t) + A(t) I_{t_{0}}^{α} y(t) + f(t),  (6)

Indeed: let x(t) be a solution of (5) and take y(t) = x′(t) ⇒ x(t) = x^{0} + I y(t), then

y(t) = A(t) \frac{d}{dt} I_{t_{0}}^{α} (x^{0} + I y(t)) + f(t)
= x^{0} \frac{(t − t_{0})^{α − 1}}{Γ(α)} A(t) + A(t) \frac{d}{dt} I I_{t_{0}}^{α} y(t) + f(t)
= x^{0} \frac{(t − t_{0})^{α − 1}}{Γ(α)} A(t) + A(t) I_{t_{0}}^{α} y(t) + f(t).

Conversely Let y(t) be a solution of (6) and take y(t) = x′(t) ⇒ x(t) = x^{0} + I y(t) and x(t_{0}) = x^{0}, then

x′(t) = x^{0} \frac{(t − t_{0})^{α − 1}}{Γ(α)} A(t) + A(t) I_{t_{0}}^{α} x′(t) + f(t)
= A(t) \frac{d}{dt} I_{t_{0}}^{α} x(t) + f(t).

Which proves the equivalence.

Now define the operator F : B → B by

F y(t) = x^{0} \frac{(t − t_{0})^{α − 1}}{Γ(α)} A(t) + A(t) I_{t_{0}}^{α} y(t) + f(t).  (7)

Let y_{i}, z_{i} ∈ B, then

F y_{i}(t) − F z_{i}(t) = a_{ij}(t) I_{t_{0}}^{α} (y_{j}(t) − z_{j}(t)),
e^{−Nt} |F y_{i}(t) − F z_{i}(t)| ≤ e^{−Nt} |a_{ij}(t)| ∫_{t_{0}}^{t} \frac{(t − s)^{α − 1}}{Γ(α)} |y_{j}(s) − z_{j}(s)| ds,
∫_{t_{0}}^{t} e^{−Ns}|F y_{i}(s) − F z_{i}(s)| ds ≤ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Ns} ∫_{t_{0}}^{s} \frac{(s − θ)^{α − 1}}{Γ(α)} |y_{j}(θ) − z_{j}(θ)| dθ ds
≤ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Nθ}|y_{j}(θ) − z_{j}(θ)| ∫_{θ}^{t} e^{−N(s−θ)} \frac{(s − θ)^{α−1}}{Γ(α)} ds dθ
≤ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Nθ}|y_{j}(θ) − z_{j}(θ)| ∫_{0}^{N(t−θ)} e^{−u} \frac{u^{α−1}}{N^{α−1}Γ(α)} \frac{du}{N} dθ
≤ \frac{||a_{ij}^{∗}||}{N^{α}} ||y_{j} − z_{j}||_{1},

therefore

||F y_{i} − F z_{i}||_{1} ≤ \frac{||a_{ij}^{∗}||}{N^{α}} ||y_{j} − z_{j}||_{1},
∑_{i=1}^{n} ||F y_{i} − F z_{i}||_{1} ≤ ∑_{i=1}^{n} \frac{||a_{ij}^{∗}||}{N^{α}} ||y_{j} − z_{j}||_{1},
||F y − F z||_{1} ≤ \frac{||A^{∗}||}{N^{α}} ||y − z||_{1}.

Now choose N large enough such that ||A^{∗}|| < N^{α}, then we get

||F y − F z||_{1} < ||y − z||_{1},

therefore the map F : B → B is contraction and (7) has a unique fixed point y ∈ B[t_{0}, T], therefore we deduce that the problem (5) has a unique solution x ∈ AC^{∗}[t_{0}, T].

3 Stability of non-autonomous systems

In this section we study the stability of the solution of the initial-value problems (2) and (5).

Theorem 3.1

The solution of the initial-value problem (2) is uniformly stable Proof. Let y(t) be a solution of

y(t) = x^{0} \frac{d}{dt} I_{t_{0}}^{α} A(t) + \frac{d}{dt} I_{t_{0}}^{α} A(t) I y(t) + \frac{d}{dt} I_{t_{0}}^{α} f(t)
= A(t_{0}) x^{0} \frac{(t − t_{0})^{α−1}}{Γ(α)} + x^{0} I_{t_{0}}^{α} A′(t) + I_{t_{0}}^{α} A′(t) I y(t) + I_{t_{0}}^{α} A(t) y(t) + \frac{d}{dt} I_{t_{0}}^{α} f(t),

and let ỹ(t) be a solution of the above linear system such that ỹ(t_{0}) = x̃^{0}, then

y_{i}(t) − ỹ_{i}(t) = (x_{j}^{0} − x̃_{j}^{0}) a_{ij}(t_{0}) \frac{(t − t_{0})^{α−1}}{Γ(α)} + (x_{j}^{0} − x̃_{j}^{0}) I_{t_{0}}^{α} a′_{ij}(t)
+ I_{t_{0}}^{α} a′_{ij}(t) I (y_{j}(t) − ỹ_{j}(t)) + I_{t_{0}}^{α} a_{ij}(t) (y_{j}(t) − ỹ_{j}(t)),
e^{−Nt} |y_{i}(t) − ỹ_{i}(t)| ≤ e^{−Nt} |x_{j}^{0} − x̃_{j}^{0}| |a_{ij}(t_{0})| \frac{(t − t_{0})^{α−1}}{Γ(α)}
+ e^{−Nt} |x_{j}^{0} − x̃_{j}^{0}| ∫_{t_{0}}^{t} \frac{(t − s)^{α−1}}{Γ(α)} |a′_{ij}(s)| ds
+ e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α−1}}{Γ(α)} |a′_{ij}(s)| ∫_{t_{0}}^{s} |y_{j}(θ) − ỹ_{j}(θ)| dθ ds
+ e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α−1}}{Γ(α)} |a_{ij}(s)| |y_{j}(s) − ỹ_{j}(s)| ds

≤ ||a_{ij}^{∗}|| e^{−Nt} |x_{j}^{0} − x̃_{j}^{0}| \frac{(t − t_{0})^{α−1}}{Γ(α)} + ||ã_{ij}|| e^{−Nt} |x_{j}^{0} − x̃_{j}^{0}| \frac{(t − t_{0})^{α}}{Γ(1 + α)}
+ ||ã_{ij}|| e^{−Nt} ∫_{t_{0}}^{t} |y_{j}(θ) − ỹ_{j}(θ)| ∫_{θ}^{t} \frac{(t − s)^{α−1}}{Γ(α)} ds dθ
+ ||a_{ij}^{∗}|| e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α−1}}{Γ(α)} |y_{j}(s) − ỹ_{j}(s)| ds
≤ ||a_{ij}^{∗}|| e^{−Nt} |x_{j}^{0} − x̃_{j}^{0}| \frac{(t − t_{0})^{α−1}}{Γ(α)} + ||ã_{ij}|| e^{−Nt} |x_{j}^{0} − x̃_{j}^{0}| \frac{(t − t_{0})^{α}}{Γ(1 + α)}
+ ||ã_{ij}|| e^{−Nt} ∫_{t_{0}}^{t} |y_{j}(θ) − ỹ_{j}(θ)| \frac{(t − θ)^{α}}{Γ(1 + α)} dθ
+ ||a_{ij}^{∗}|| e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α−1}}{Γ(α)} |y_{j}(s) − ỹ_{j}(s)| ds

and

∫_{t_{0}}^{t} e^{−Ns} |y_{i}(s) − ỹ_{i}(s)| ds ≤ ||a_{ij}^{∗}|| e^{−Nt_{0}} |x_{j}^{0} − x̃_{j}^{0}| ∫_{t_{0}}^{t} e^{−N(s−t_{0})} \frac{(s − t_{0})^{α−1}}{Γ(α)} ds
+ ||ã_{ij}|| e^{−Nt_{0}} |x_{j}^{0} − x̃_{j}^{0}| ∫_{t_{0}}^{t} e^{−N(s−t_{0})} \frac{(s − t_{0})^{α}}{Γ(1 + α)} ds
+ ||ã_{ij}|| ∫_{t_{0}}^{t} e^{−Ns} ∫_{t_{0}}^{s} |y_{j}(θ) − ỹ_{j}(θ)| \frac{(s − θ)^{α}}{Γ(1 + α)} dθ ds
+ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Ns} ∫_{t_{0}}^{s} \frac{(s − θ)^{α−1}}{Γ(α)} |y_{j}(θ) − ỹ_{j}(θ)| dθ ds
≤ ||a_{ij}^{∗}|| ||x_{j}^{0} − x̃_{j}^{0}||_{2} ∫_{0}^{N(t−t_{0})} e^{−u} \frac{u^{α−1}}{N^{α−1} Γ(α)} \frac{du}{N}
+ ||ã_{ij}|| ||x_{j}^{0} − x̃_{j}^{0}||_{2} ∫_{0}^{N(t−t_{0})} e^{−u} \frac{u^{α}}{N^{α} Γ(1 + α)} \frac{du}{N}
+ ||ã_{ij}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − ỹ_{j}(θ)| ∫_{θ}^{t} e^{−N(s−θ)} \frac{(s − θ)^{α}}{Γ(1 + α)} ds dθ
+ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − ỹ_{j}(θ)| ∫_{θ}^{t} e^{−N(s−θ)} \frac{(s − θ)^{α−1}}{Γ(α)} ds dθ
≤ \frac{||a_{ij}^{∗}||}{N^{α}} ||x_{j}^{0} − x̃_{j}^{0}||_{2} + \frac{||ã_{ij}||}{N^{1+α}} ||x_{j}^{0} − x̃_{j}^{0}||_{2}
+ ||ã_{ij}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − ỹ_{j}(θ)| ∫_{0}^{N(t−θ)} e^{−u} \frac{u^{α}}{N^{α} Γ(1 + α)} \frac{du}{N} dθ
+ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − ỹ_{j}(θ)| ∫_{0}^{N(t−θ)} e^{−u} \frac{u^{α−1}}{N^{α−1} Γ(α)} \frac{du}{N} dθ,

then

||y_{i} − ỹ_{i}||_{1} ≤ \frac{||a_{ij}^{∗}||}{N^{α}} ||x_{j}^{0} − x̃_{j}^{0}||_{2} + \frac{||ã_{ij}||}{N^{1+α}} ||x_{j}^{0} − x̃_{j}^{0}||_{2}

+ \frac{||ã_{ij}||}{N^{1+α}} ||y_{j} − ỹ_{j}||_{1} + \frac{||a_{ij}^{∗}||}{N^{α}} ||y_{j} − ỹ_{j}||_{1},
(1 − \frac{||ã_{ij}||}{N^{1+α}} − \frac{||a_{ij}^{∗}||}{N^{α}}) ||y_{j} − ỹ_{j}||_{1} ≤ (\frac{||a_{ij}^{∗}||}{N^{α}} + \frac{||ã_{ij}||}{N^{1+α}} ) ||x_{j}^{0} − x̃_{j}^{0}||_{2}

⇒ ||y_{i} − ỹ_{i}||_{1} ≤ (1 − \frac{||ã_{ij}||}{N^{1+α}} − \frac{||a_{ij}^{∗}||}{N^{α}})^{−1} (\frac{||a_{ij}^{∗}||}{N^{α}} + \frac{||ã_{ij}||}{N^{1+α}} ) ||x_{j}^{0} − x̃_{j}^{0}||_{2}.

Now, since

x(t) = x^{0} + I y(t),

then

x_{i}(t) − x̃_{i}(t) = x_{i}^{0} − x̃_{i}^{0} + ∫_{t_{0}}^{t} (y_{i}(s) − ỹ_{i}(s)) ds,
e^{−Nt}|x_{i}(t) − x̃_{i}(t)| ≤ e^{−Nt} |x_{i}^{0} − x̃_{i}^{0}| + e^{−Nt} ∫_{t_{0}}^{t} |y_{i}(s) − ỹ_{i}(s)| ds
≤ e^{−Nt_{0}} |x_{i}^{0} − x̃_{i}^{0}| + e^{−Nt} ∫_{t_{0}}^{t} e^{Ns} e^{−Ns} |y_{i}(s) − ỹ_{i}(s)| ds,
||x_{i} − x̃_{i}||_{2} ≤ ||x_{i}^{0} − x̃_{i}^{0}||_{2} + ||y_{i} − ỹ_{i}||_{1}
≤ ||x_{i}^{0} − x̃_{i}^{0}||_{2}
+ (1 − \frac{||ã_{ij}||}{N^{1+α}} − \frac{||a_{ij}^{∗}||}{N^{α}})^{−1} (\frac{||a_{ij}^{∗}||}{N^{α}} + \frac{||ã_{ij}||}{N^{1+α}} ) ||x_{j}^{0} − x̃_{j}^{0}||_{2}

≤ (1 − \frac{||ã_{ij}||}{N^{1+α}} − \frac{||a_{ij}^{∗}||}{N^{α}})^{−1} ||x_{j}^{0} − x̃_{j}^{0}||_{2}.

Therefore

∑_{i=1}^{n} ||x_{i} − x̃_{i}||_{2} ≤ ∑_{i=1}^{n} (1 − \frac{||ã_{ij}||}{N^{1+α}} − \frac{||a_{ij}^{∗}||}{N^{α}})^{−1} ||x_{j}^{0} − x̃_{j}^{0}||_{2},

||x − x̃||_{2} ≤ (1 − \frac{||Ã||}{N^{1+α}} − \frac{||A^{∗}||}{N^{α}})^{−1} ||x^{0} − x̃^{0}||_{2}

Therefore, if ||x^{0} − x̃^{0}||_{2} < δ(ε), then ||x − x̃||_{2} < ε, which complete the proof of the theorem.

Theorem 3.2

The solution of the initial-value problem (5) is uniformly stable. Proof. Let y(t) be a solution of

y(t) = x^{0} \frac{(t − t_{0})^{α − 1}}{Γ(α)} A(t) + A(t) I_{t_{0}}^{α} y(t) + f(t)

and let ỹ(t) be a solution of the above linear system such that ỹ(t_{0}) = x̃^{0}, then

y_{i}(t) − ỹ_{i}(t) = (x_{j}^{0} − x̃_{j}^{0}) \frac{(t − t_{0})^{α − 1}}{Γ(α)} a_{ij}(t) + a_{ij}(t) I_{t_{0}}^{α} (y_{j}(t) − ỹ_{j}(t)),
e^{−Nt} |y_{i}(t) − ỹ_{i}(t)| ≤ e^{−Nt} |x_{j}^{0} − x̃_{j}^{0}| \frac{(t − t_{0})^{α−1}}{Γ(α)} | |a_{ij}(t)|
+ |a_{ij}(t)| e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α−1}}{Γ(α)} |y_{j}(s) − ỹ_{j}(s)| ds
≤ ||a_{ij}^{∗}|| e^{−Nt} |x_{j}^{0} − x̃_{j}^{0}| \frac{(t − t_{0})^{α−1}}{Γ(α)}
+ ||a_{ij}^{∗}|| e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α−1}}{Γ(α)} |y_{j}(s) − ỹ_{j}(s)| ds,
∫_{t_{0}}^{t} e^{−Ns} |y_{j}(s) − ỹ_{j}(s)| ds ≤ ||a_{ij}^{∗}|| e^{−Nt_{0}} |x_{j}^{0} − x̃_{j}^{0}| ∫_{t_{0}}^{t} e^{−N(s−t_{0})} \frac{(s − t_{0})^{α−1}}{Γ(α)} ds
+ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Ns} ∫_{t_{0}}^{s} \frac{(s − θ)^{α−1}}{Γ(α)} |y_{j}(θ) − ỹ_{j}(θ)| dθ ds
≤ ||a_{ij}^{∗}|| ||x_{j}^{0} − x̃_{j}^{0}||_{2} ∫_{0}^{N(t−t_{0})} e^{−u} \frac{u^{α−1}}{N^{α−1} Γ(α)} \frac{du}{N}
+ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − ỹ_{j}(θ)| ∫_{θ}^{t} e^{−N(s−θ)} \frac{(s − θ)^{α−1}}{Γ(α)} ds dθ
≤ \frac{||a_{ij}^{∗}||}{N^{α}} ||x_{j}^{0} − x̃_{j}^{0}||_{2}
+ ||a_{ij}^{∗}|| ∫_{t_{0}}^{t} e^{−Nθ} |y_{j}(θ) − ỹ_{j}(θ)| ∫_{0}^{N(t−θ)} e^{−u} \frac{u^{α−1}}{N^{α−1} Γ(α)} \frac{du}{N} dθ
≤ \frac{||a_{ij}^{∗}||}{N^{α}} ||x_{j}^{0} − x̃_{j}^{0}||_{2} + \frac{||a_{ij}^{∗}||}{N^{α}} ||y_{j} − ỹ_{j}||_{1},

then

||y_{i} − ỹ_{i}||_{1} ≤ \frac{||a_{ij}^{∗}||}{N^{α}} ||x_{j}^{0} − x̃_{j}^{0}||_{2} + \frac{||a_{ij}^{∗}||}{N^{α}} ||y_{j} − ỹ_{j}||_{1},
(1 − \frac{||a_{ij}^{∗}||}{N^{α}}) ||y_{j} − ỹ_{j}||_{1} ≤ \frac{||a_{ij}^{∗}||}{N^{α}} ||x_{j}^{0} − x̃_{j}^{0}||_{2}

⇒ ||y_{i} − ỹ_{i}||_{1} ≤ \frac{||a_{ij}^{∗}||}{N^{α}} (1 − \frac{||a_{ij}^{∗}||}{N^{α}})^{−1} ||x_{j}^{0} − x̃_{j}^{0}||_{2}

≤ (\frac{||a_{ij}^{∗}||}{N^{α} − ||a_{ij}^{∗}||}) ||x_{j}^{0} − x̃_{j}^{0}||_{2}.

Now, since

x(t) = x^{0} + I y(t),

then

||x_{i} − x̃_{i}||_{2} ≤ ||x_{i}^{0} − x̃_{i}^{0}||_{2} + ||y_{i} − ỹ_{i}||_{1}

≤ ||x_{i}^{0} − x̃_{i}^{0}||_{2} + (\frac{||a_{ij}^{∗}||}{N^{α} − ||a_{ij}^{∗}||}) ||x_{j}^{0} − x̃_{j}^{0}||_{2}

≤ (\frac{N^{α}}{N^{α} − ||a_{ij}^{∗}||}) ||x_{j}^{0} − x̃_{j}^{0}||_{2}.

Therefore

∑_{i=1}^{n} ||x_{i} − x̃_{i}||_{2} ≤ ∑_{i=1}^{n} (\frac{N^{α}}{N^{α} − ||a_{ij}^{∗}||}) ||x_{j}^{0} − x̃_{j}^{0}||_{2},
||x − x̃||_{2} ≤ (\frac{N^{α}}{N^{α} − ||A^{∗}||}) ||x^{0} − x̃^{0}||_{2}

Therefore, if ||x^{0} − x̃^{0}||_{2} < δ(ε), then ||x − x̃||_{2} < ε, which complete the proof of the theorem.

4 Autonomous systems

Now we study the problems:

D_{t_{0}}^{α} x(t) = A x(t), α ∈ (0, 1], x(t_{0}) = x^{0}  (8)

and

x′(t) = A \frac{d}{dt} I_{t_{0}}^{α} x(t), α ∈ (0, 1], x(t_{0}) = x^{0}  (9)

which are the special cases of the initial-value problems (2) and (5) when A(t) = A, where A is a real constant matrix and f(t) is the zero vector.

Theorem 4.1

The solution of the initial-value problem (8) or (9) is given by the formula

x(t) = ∑_{k=0}^{∞} (A I_{t_{0}}^{α})^{k} x^{0}

= ∑_{k=0}^{∞} \frac{A^{k} (t − t_{0})^{kα}}{Γ(1 + kα)} x^{0}

Proof. Firstly we prove that the two problems are equivalent to each other, indeed: Let x(t) be a solution of (8), then

I_{t_{0}}^{1 − α} \frac{dx}{dt} = A x(t),

operating by I_{t_{0}}^{α} on both sides of the last relation, we get

x(t) = x^{0} + A I_{t_{0}}^{α} x(t),

differentiating both sides, we obtain (9) and when t = t_{0} , we obtain x(t_{0}) = x^{0}. Conversely let x(t) be a solution of (9), operating by I_{t_{0}}^{1 − α} on both sides of it, we get

I_{t_{0}}^{1 − α} \frac{dx}{dt} = A I_{t_{0}}^{1 − α} (x^{0} \frac{(t − t_{0})^{α−1}}{Γ(α)} + I_{t_{0}}^{α} x′(t))
⇒ D_{t_{0}}^{α} x(t) = A (x^{0} + x(t) − x^{0}) = A x(t).

Now since

e^{−Nt} |a_{ij} I_{t_{0}}^{α} x_{j}(t)| ≤ |a_{ij}| e^{−Nt} ∫_{t_{0}}^{t} \frac{(t − s)^{α − 1}}{Γ(α)} |x_{j}(s)| ds
≤ |a_{ij}| ∫_{t_{0}}^{t} e^{−N(t − s)} \frac{(t − s)^{α − 1}}{Γ(α)} e^{−Ns} |x_{j}(s)| ds,
||a_{ij} I_{t_{0}}^{α} x_{j}||_{2} ≤ |a_{ij}| ||x_{j}||_{2} ∫_{0}^{N(t−t_{0})} e^{−u} \frac{u^{α−1}}{N^{α−1} Γ(α)} \frac{du}{N}
≤ \frac{|a_{ij}|}{N^{α}} ||x_{j}||_{2},

then

∑_{i=1}^{n} ||a_{ij} I_{t_{0}}^{α} x_{j}||_{2} ≤ \frac{1}{N^{α}} ∑_{i=1}^{n} |a_{ij}| ||x_{j}||_{2},  i = 1, 2, ..., n,
||A I_{t_{0}}^{α} x||_{2} ≤ \frac{||A||}{N^{α}} ||x||_{2} < ||x||_{2},

where ||A|| < N^{α}, it follows that ||A I_{t_{0}}^{α}||_{2} < 1, then from Neumann expansion (see [3]) we complete the proof.

Theorem 4.2

If A is a real constant matrix with the characteristic roots all having negative real parts, then the solution of the initial-value problem (8) (or (9)) is uniformly asymptotically stable. Proof.

|x_{i}(t) − x̃_{i}(t)| = |x_{i}(t) − e^{a_{ij}(t − t_{0})} x_{j}^{0} − x̃_{i}(t) + e^{a_{ij}(t − t_{0})} x̃_{j}^{0}
+ e^{a_{ij}(t − t_{0})} x_{j}^{0} − e^{a_{ij}(t − t_{0})} x̃_{j}^{0}|
= |(x_{i}(t) − e^{a_{ij}(t − t_{0})} x_{j}^{0}) − (x̃_{i}(t) − e^{a_{ij}(t − t_{0})} x̃_{j}^{0})
+ e^{a_{ij}(t − t_{0})} (x_{j}^{0} − x̃_{j}^{0})|
= |(∑_{k=0}^{∞} \frac{a_{ij}^{k} (t − t_{0})^{kα}}{Γ(1 + kα)} x_{j}^{0} − e^{a_{ij}(t − t_{0})} x_{j}^{0})

− (∑_{k=0}^{∞} \frac{a_{ij}^{k} (t − t_{0})^{kα}}{Γ(1 + kα)} x̃_{j}^{0} − e^{a_{ij}(t−t_{0})} x̃_{j}^{0}) + e^{a_{ij}(t−t_{0})} (x_{j}^{0} − x̃_{j}^{0})|

≤ | ∑_{k=0}^{∞} \frac{a_{ij}^{k} (t − t_{0})^{kα}}{Γ(1 + kα)} − ∑_{k=0}^{∞} \frac{a_{ij}^{k} (t − t_{0})^{k}}{Γ(1 + k)} | |x_{j}^{0} − x̃_{j}^{0}|

+ e^{a_{ij}(t − t_{0})} |x_{j}^{0} − x̃_{j}^{0}|
= | − ∑_{k=0}^{∞} \frac{a_{ij}^{k}}{Γ(1 + k)} (1 − \frac{Γ(1 + k)}{Γ(1 + kα)} (t − t_{0})^{kα−k}) (t − t_{0})^{k}| |x_{j}^{0} − x̃_{j}^{0}|
+ e^{a_{ij}(t − t_{0})} |x_{j}^{0} − x̃_{j}^{0}|.

Let k α = k − β, α ∈ (0, 1], β > 0, then

|x_{i}(t) − x̃_{i}(t)| ≤ | − ∑_{k=0}^{∞} \frac{a_{ij}^{k}}{Γ(1 + k)} (1 − \frac{Γ(1 + k)}{Γ(1 + k − β)} (t − t_{0})^{−β}) (t − t_{0})^{k}| |x_{j}^{0} − x̃_{j}^{0}|
+ e^{a_{ij}(t − t_{0})} |x_{j}^{0} − x̃_{j}^{0}|
≤ | ∑_{k=0}^{∞} \frac{a_{ij}^{k} (t − t_{0})^{k}}{Γ(1 + k)} | |x_{j}^{0} − x̃_{j}^{0} | + e^{a_{ij}(t − t_{0})} |x_{j}^{0} − x̃_{j}^{0}|
= e^{a_{ij}(t − t_{0})} |x_{j}^{0} − x̃_{j}^{0}| + e^{a_{ij}(t − t_{0})} |x_{j}^{0} − x̃_{j}^{0}|
= 2 e^{a_{ij}(t − t_{0})} |x_{j}^{0} − x̃_{j}^{0}|.

Since A is a real constant matrix with the characteristic roots all having negative real parts, then there exists positive constants K and σ such that

e^{At} ≤ K e^{−σt},

therefore

e^{−Nt} |x_{i}(t) − x̃_{i}(t)| ≤ 2 K e^{−σ (t − t_{0})} e^{−Nt} |x_{i}^{0} − x̃_{i}^{0}|
≤ 2 K e^{−σ (t − t_{0})} e^{−Nt_{0}} |x_{i}^{0} − x̃_{i}^{0}|,
||x_{i} − x̃_{i}||_{2} ≤ 2 K e^{−σ (t − t_{0})} ||x_{i}^{0} − x̃_{i}^{0}||_{2},
∑_{i=1}^{n} ||x_{i} − x̃_{i}||_{2} ≤ 2 K e^{−σ (t − t_{0})} ∑_{i=1}^{n} ||x_{i}^{0} − x̃_{i}^{0}||_{2},
||x − x̃||_{2} ≤ 2 K e^{−σ (t − t_{0})} ||x^{0} − x̃^{0}||_{2}.

Therefore, we deduce that the solution is uniformly asymptotically stable.

References

[1] Corduneanu, C. Principles of Differential and Integral equations, Allyn and Bacon, Inc., Boston, Massachusetts (1971).

[2] Miller, K. S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York (1993).

[3] Oden, J. T. Applied Functional Analysis, Prentice - Hall, Inc. Englwood Cliffs, New Jersey (1979).

[4] Podlubny, I. and EL-Sayed, A. M. A. On two definitions of fractional calculus, Preprint UEF 03-96 (ISBN 80-7099-252-2), Slovak Academy of Science-Institute of Experimental phys. (1996).

[5] Podlubny, I. Fractional Differential Equations, Acad. press, San Diego-New York-London (1999).

[6] Samko, S., Kilbas, A. and Marichev, O. L. Fractional Integrals and Derivatives, Gordon and Breach Science Publisher, (1993).

(Received December 12, 2006)